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Skill: polynomial-operations

Polynomial Operations

Add, subtract, and multiply polynomials with confidence. Combine like terms, distribute every term to every term, and master FOIL plus the two special products that show up everywhere.

1 Understand

The core idea in plain language.

What it is. A polynomial is a sum of terms like 3x2 − 5x + 7. Adding and subtracting polynomials just means combining like terms (same variable, same exponent). Multiplying means distributing every term to every term — FOIL for two binomials, or the box/area method for bigger ones. Watch for special products: (a+b)2 = a2 + 2ab + b2 and (a+b)(a−b) = a2 − b2.

Why it matters. Polynomials are the vocabulary of algebra — every equation, function, and factoring problem speaks it. Fluency here makes factoring, quadratics, and graphing dramatically easier.

Where it is used. Area formulas with variables · projectile motion (quadratic functions) · computer graphics curves.

2 See It

Diagrams that make the idea visual.

Multiplication is area

Each term of the first binomial stretches across each term of the second — the box catches every piece so none go missing.

The four regions are x2, 3x, 5x, and 15. Adding them: x2 + 3x + 5x + 15 = x2 + 8x + 15.

x 3 x 5 x² 3x 5x 15 (x + 3)(x + 5) = x² + 8x + 15

Multiplication is area. Each term of the first binomial stretches across each term of the second — the box catches every piece so none go missing.

Only identical twins combine

Like terms share the same variable and the same exponent. 3x2 and −x2 merge; 2x cannot join them — different exponent, different family.

3x2 − x2 = 2x2; 2x − 7x = −5x; −5 + 4 = −1.

3x² −x² 2x −7x −5 +4 2x² − 5x − 1

Only identical twins combine. 3x² and −x² merge; 2x cannot join them — different exponent, different family.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Add: (3x² + 2x − 5) + (x² − 7x + 4)
  1. Group like terms: (3x2 + x2) + (2x − 7x) + (−5 + 4).
  2. Combine each group: 4x2, −5x, −1.
(3x² + 2x − 5) + (x² − 7x + 4) = 4x² − 5x − 1

Check: Degrees and term counts look right; no like terms remain uncombined.

Example 2 Subtract: (5x³ − 2x) − (2x³ + x² − 9)
  1. Distribute the minus to every term of the second polynomial first: 5x3 − 2x − 2x3 − x2 + 9.
  2. Group and combine: (5x3 − 2x3) − x2 − 2x + 9.
(5x³ − 2x) − (2x³ + x² − 9) = 3x³ − x² − 2x + 9

Check: The −9 became +9 and +x² became −x² — the minus hit everything.

Example 3 Multiply: (x + 4)(x − 6)
  1. FOIL — First, Outer, Inner, Last.
  2. x2 − 6x + 4x − 24, then combine the middle terms.
(x + 4)(x − 6) = x² − 2x − 24

Check: Substitute x = 0: (4)(−6) = −24 and 0 − 0 − 24 = −24.

Example 4 Special product: (2x + 3)²
  1. Use (a+b)2 = a2 + 2ab + b2 with a = 2x, b = 3.
  2. (2x)2 + 2(2x)(3) + 32 = 4x2 + 12x + 9.
(2x + 3)² = 4x² + 12x + 9

Check: (2x+3)(2x+3) = 4x² + 6x + 6x + 9.

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: (x + 3)² = x² + 9
Wrong

Squaring each term and skipping the middle: (x + 3)2 = x2 + 9.

Right

x2 + 6x + 9 — the 2ab term is not optional.

Fix: (x+3)2 means (x+3)(x+3). FOIL it out at least until the pattern is automatic.

Mistake 2: Sign errors when subtracting
Wrong

(4x2 − x + 3) − (x2 + 2x − 8) = “3x2 + x − 5″ — the minus only hit the first term.

Right

3x2 − 3x + 11 — the minus distributes to every term.

Fix: Rewrite subtraction as “plus the opposite” first, then combine.

Mistake 3: 2x · 3x = 6x
Wrong

Multiplying coefficients but forgetting exponents: 2x · 3x = 6x.

Right

6x2 — coefficients multiply AND exponents add.

Fix: Numbers × numbers, letters × letters. x·x = x2, always.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Multiply (x − 5)(x + 2).

Answer

(x − 5)(x + 2) = x2 + 2x − 5x − 10 = x2 − 3x − 10
Check with x = 0: (−5)(2) = −10. ✓

2. Subtract: (4x² − x + 3) − (x² + 2x − 8).

Answer

4x2 − x + 3 − x2 − 2x + 8 = 3x2 − 3x + 11
The minus flipped every sign inside the second parentheses.

3. Multiply (x + 9)(x − 9).

Answer

(x + 9)(x − 9) = x2 − 81 — difference of squares: the middle terms cancel.
Check with x = 0: (9)(−9) = −81. ✓

Key Points to Remember

  • Adding/subtracting = combining like terms (same variable, same exponent).
  • Multiplying = distribute every term to every term (FOIL for binomials).
  • (a + b)2 = a2 + 2ab + b2 — the middle term is not optional.
  • (a + b)(a − b) = a2 − b2 — the middle terms cancel.
  • Subtracting a polynomial flips every sign inside the parentheses.
  • xm · xn = xm+n — exponents add when multiplying.
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